Abstract
We study scrambling, an avatar of chaos, in a weakly interacting metal in the presence of random potential disorder. It is well known that charge and heat spread via diffusion in such an interacting disordered metal. In contrast, we show within perturbation theory that chaos spreads in a ballistic fashion. The squared anticommutator of the electron-field operators inherits a light-cone-like growth, arising from an interplay of a growth (Lyapunov) exponent that scales as the inelastic electron scattering rate and a diffusive piece due to the presence of disorder. In two spatial dimensions, the Lyapunov exponent is universally related at weak coupling to the sheet resistivity. We are able to define an effective temperature-dependent butterfly velocity, a speed limit for the propagation of quantum information that is much slower than microscopic velocities such as the Fermi velocity and that is qualitatively similar to that of a quantum critical system with a dynamical critical exponent .
- Received 1 April 2017
DOI:https://doi.org/10.1103/PhysRevX.7.031047
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Recent theoretical advances have raised new questions about the way chaos manifests in complex quantum systems consisting of many interacting particles. One of the key questions is how fast the system loses memory of its initial state, or information is “scrambled,” by the chaotic motion of quantum particles. One of the most surprising developments is that there are deep connections to the physics of black holes, which are believed to be the fastest scramblers in the Universe, saturating a universal bound on quantum chaos. Our work studies the physics of scrambling and information speed limits in a paradigmatic example of a quantum many-particle system—a metal consisting of electrons with Coulomb interactions and static impurities.
Using a sophisticated real-time approach in which time effectively runs both forward and backward, we calculate the rate of growth of chaos and the speed of information spreading in these interacting disordered metals. Surprisingly, while the speed of light provides a fundamental speed limit on information spreading, these metals obey a much stricter emergent speed limit. The limiting speed is found to have a power-law dependence on temperature, and it vanishes as the temperature approaches absolute zero. Another physical consequence is that the two-dimensional version of the system must undergo a phase transition (as seen in experiments) as interactions increase; otherwise, the bound on chaos would be violated.
Besides exhibiting a new kind of low-temperature speed limit, these results may have bearing on other aspects of information dynamics, including the rate of production of entanglement and entropy. More generally, the results may guide upcoming experiments designed to map out the range of scrambling behaviors from the very slow to the very fast.